[ChrisLomont]

There is no other way. The problem is atoms combine in integer ratio amounts to form molecules. To mix things for chemistry one needs to be able to mix things in proper proportions. So having a number that trades number of atoms to something plausibly measurable, like mass, is needed.

The reason it cannot be exact for all atoms is forced on us by nature: atoms come in isotopes, each weighing slightly differently, and most common elements come in a mix of isotopes.

So picking one isotope of one element (carbon-12) as the definition for a mole that is decently representative of how chemists will use the number is a perfectly fine and useful number.

Any chemist that needs to worry about the fuzz will understand this and act accordingly. For example, carbon 22 has a mass slightly larger than 22/12 that of carbon 12 (but has short half-life). Carbon 13, which is stable, has mass slightly over 13/12 that of carbon 12, and when using it, one adjusts accordingly. And these "slightly over" phrases are also known to many digits of precision.

But nailing down the number precisely is extremely useful.